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K-theory, L^2-invariants, manifolds, groups and their interactions

Periodic Reporting for period 3 - KL2MG-interactions (K-theory, L^2-invariants, manifolds, groups and their interactions)

Reporting period: 2018-11-01 to 2020-04-30

The primary topics are the outstanding and deep conjectures of Farrell-Jones on the algebraic
K- and L-heory of group rings, of Baum-Connes on the topological K-theory of
reduced group C^*-algebras, and of Atiyah on the integrality of L^2-Betti numbers, and
further open problems such as the approximation of L^2-torsion for towers of finite
coverings, and the relation of the first L^2-Betti number, the cost and the rank
gradient of a finitely presented group.

Interactions between the classification of manifolds and invariants in algebraic K- and
L-theory have been studied since decades. L^2-invariants have been applied in
the recent years to problems about manifolds, groups and von Neumann algebras.
Much less developed and understood and not at all exploited are
interactions between K- and L-theory of group rings and L^2-invariants.

One goal is to extend Farrell-Jones Conjecture to other settings such as topological cyclic homology of
``group rings'' over the sphere spectrum, algebraic K-theory of Hecke algebras of totally disconnected groups,
the topological K-theory of Frechet group algebras, and Waldhausen's A-theory of classifying spaces of groups.
This has very deep and interesting consequences for automorphism groups of closed aspherical manifolds,
the structure of group rings,and representation theory. Recent
proofs of the Farrell-Jones Conjecture for certain classes of groups by the PI require input from
homotopy theory, geometric group theory, controlled topology and flows on metric spaces,
and can hopefully be transferred to the new situations. There is also a program
towards a proof of the Atiyah Conjecture
based on the Farrell-Jones Conjecture and ring
theory. We see a high potential for new striking applications of the Farrell-Jones
Conjecture and L^2-techniques to manifolds and groups.
Objectives I.1.a and I.1.b about the claim that via the cyclotomic trace one can prove the rational
injectivity of the K-theoretic Farrell-Jones assembly map have been completely settled and resulted in the
papers [8,9]. Actually the results of the second paper go even beyond the claim of the objectives since there
it is for instance shown that the Farrell-Jones Conjecture does not hold for topological cyclic homology.

The claims appearing in Objective I.4 about the computation of Waldhausen's non-connective A-theory for
aspherical spaces have been proved in the paper [2]. There are several sequel objectives we want to attack
using this result.

Objective III.1 about block fibering manifolds has been completely solved in the paper [3].

A lot of progress including and actually going meanwhile beyond the original Objective III.3 about identifying
THurston norms and polytopes with generalised L^2-torsion invariants has been made in the papers [1,4,5,6].

A survey on the status of the deep problems occurring in Objectives~II.1 and II.2 is given in [7].

[1] J. Dubois, S. Friedl, and W. Lück.
The L^2-Alexander torsion of 3-manifolds.
J. Topol., 9(3):889--926, 2016.

[2] N.-E. Enkelmann, W. Lück, M. Pieper, M. Ullmann, and C. Winges.
On the Farrell-Jones conjecture for Waldhausen's A-theory.
to appear in Geometry and Topology.

[3] T. Farrell, W. Lück, and W. Steimle.
Approximately fibering a manifold over an aspherical one.
Math. Ann., 370(1-2):669--726, 2018.

[4] S. Friedl and W. Lück.
The L^2-torsion function and the Thurston norm of 3-manifolds.
to appear in Commentarii Mathematici Helvetici.

[5] S. Friedl and W. Lück.
Universal L^2-torsion, polytopes and applications to 3-manifolds.
Proc. Lond. Math. Soc. (3), 114(6):1114--1151, 2017.

[6] P. Linnell and W. Lück.
Localization, Whitehead groups and the Atiyah conjecture.
Annals of K-Theory, 3(1):33--53, 2018.

[7] W. Lück.
Approximating L^2-invariants by their classical counterparts.
EMS Surv. Math. Sci., 3(2):269--344, 2016.

[8] W. Lück, H. Reich, J. Rognes, and M. Varisco.
Algebraic K-theory of group rings and the cyclotomic trace map.
Adv. Math., 304:930--1020, 2017.

[9] W. Lück, H. Reich, J. Rognes, and M. Varisco.
Assembly maps for topological cyclic homology of group algebras.
to appear in Crelle, 2016.
The result of the paper [2] will be the basis for several sequel objectives we want to attack.

A lot of progress including and actually going meanwhile beyond the original Objective III.3 about identifying
THurston norms and polytopes with generalised L^2-torsion invariants has been made in the papers [1,4,5,6].
This project has already stimulated other reproach activities by other researchers which use the new invariants and
results which we have achieved so far and apply them for instance to group theory.